In an abstract framework, we study local convergence properties of Newton's method for a sequence of generalized equations which models a discretized variational inequality. We identify conditions under which the method is locally quadratically convergent, uniformly in the discretization. Moreover, we show that the distance between the Newton sequence for the continuous problem and the Newton sequence for the discretized problem is bounded by the norm of a residual. As an application, we present mesh-independence results for an optimal control problem with control constraints.